{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:LKU3DIMES3WZTYVAPX6YUFA4TT","short_pith_number":"pith:LKU3DIME","schema_version":"1.0","canonical_sha256":"5aa9b1a18496ed99e2a07dfd8a141c9cda143f357703f0476f23dd9fa5af106c","source":{"kind":"arxiv","id":"1003.3552","version":2},"attestation_state":"computed","paper":{"title":"Non-Abelian Discrete Symmetries in Particle Physics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-ph"],"primary_cat":"hep-th","authors_text":"Hajime Ishimori, Hiroshi Ohki, Hiroshi Okada, Morimitsu Tanimoto, Tatsuo Kobayashi, Yusuke Shimizu","submitted_at":"2010-03-18T11:30:06Z","abstract_excerpt":"We review pedagogically non-Abelian discrete groups, which play an important role in the particle physics. We show group-theoretical aspects for many concrete groups, such as representations, their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discussed them explicitly for $S_N$, $A_N$, $T'$, $D_N$, $Q_N$, $\\Sigma(2N^2)$, $\\Delta(3N^2)$, $T_7$, $\\Sigma(3N^3)$ and $\\Delta(6N^2)$, which have been  applied for model building in the particle physics. We also present typical flavor models by"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1003.3552","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2010-03-18T11:30:06Z","cross_cats_sorted":["hep-ph"],"title_canon_sha256":"a29f12ee64250ddb449f1c0bd6278279e8d554908ae96513b23c12c0a1c80fdd","abstract_canon_sha256":"b11fa34d145c1820725e331a12e2701ea1d9b3f001a244178beb118c6aabd9fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:36.751817Z","signature_b64":"PPCEEu0fBVRoWS9JbWr4atU2GldlG1FZJniQCbnU3UV3geO47b4Bz3DGGpXrAcmU/QMQzPCw9h/tv1Xsp5JfAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5aa9b1a18496ed99e2a07dfd8a141c9cda143f357703f0476f23dd9fa5af106c","last_reissued_at":"2026-05-18T02:24:36.751048Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:36.751048Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-Abelian Discrete Symmetries in Particle Physics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-ph"],"primary_cat":"hep-th","authors_text":"Hajime Ishimori, Hiroshi Ohki, Hiroshi Okada, Morimitsu Tanimoto, Tatsuo Kobayashi, Yusuke Shimizu","submitted_at":"2010-03-18T11:30:06Z","abstract_excerpt":"We review pedagogically non-Abelian discrete groups, which play an important role in the particle physics. We show group-theoretical aspects for many concrete groups, such as representations, their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discussed them explicitly for $S_N$, $A_N$, $T'$, $D_N$, $Q_N$, $\\Sigma(2N^2)$, $\\Delta(3N^2)$, $T_7$, $\\Sigma(3N^3)$ and $\\Delta(6N^2)$, which have been  applied for model building in the particle physics. We also present typical flavor models by"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3552","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1003.3552","created_at":"2026-05-18T02:24:36.751167+00:00"},{"alias_kind":"arxiv_version","alias_value":"1003.3552v2","created_at":"2026-05-18T02:24:36.751167+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.3552","created_at":"2026-05-18T02:24:36.751167+00:00"},{"alias_kind":"pith_short_12","alias_value":"LKU3DIMES3WZ","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"LKU3DIMES3WZTYVA","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"LKU3DIME","created_at":"2026-05-18T12:26:10.704358+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":15,"internal_anchor_count":12,"sample":[{"citing_arxiv_id":"2607.07853","citing_title":"Neutrino Masses and Dark Matter Stability in 3HDMs with Minimal Non-Abelian Discrete Symmetries","ref_index":17,"is_internal_anchor":true},{"citing_arxiv_id":"2607.06102","citing_title":"Radiative Breaking of Two-Zero Neutrino Mass Minors: Revisiting the $\\mathrm{U}(1)_{L_\\mu-L_\\tau}$ Model","ref_index":7,"is_internal_anchor":true},{"citing_arxiv_id":"2606.25027","citing_title":"Fermion mass relations in one-parameter modular models","ref_index":4,"is_internal_anchor":true},{"citing_arxiv_id":"2606.24277","citing_title":"Non-holomorphic $S^{\\prime}_{4}$ modular symmetry for leptons and leptogenesis","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2605.28141","citing_title":"More about modular symmetries and non-invertible properties in magnetized compactifications","ref_index":42,"is_internal_anchor":true},{"citing_arxiv_id":"1907.09172","citing_title":"$A_4$ flavor symmetric model in SUSY SU(5) GUT","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2303.02358","citing_title":"Equal Majorana Phases from a Minimal and Predictive Neutrino Texture","ref_index":37,"is_internal_anchor":true},{"citing_arxiv_id":"2503.21432","citing_title":"Exploring the flavor structure of leptons via diffusion models","ref_index":3,"is_internal_anchor":true},{"citing_arxiv_id":"2506.08080","citing_title":"Towards AI-assisted Neutrino Flavor Theory Design","ref_index":18,"is_internal_anchor":true},{"citing_arxiv_id":"2509.10373","citing_title":"The Seesaw Evaded Modular Dirac Framework","ref_index":1,"is_internal_anchor":true},{"citing_arxiv_id":"2602.11277","citing_title":"Fate of $\\theta_{12}$ under $\\mu-\\tau$ Reflection Symmetry in Light of the First JUNO Results","ref_index":62,"is_internal_anchor":true},{"citing_arxiv_id":"2603.15455","citing_title":"Why Quarks and Leptons Demand Different Symmetries: A Systematic $Z_3$ Froggatt-Nielsen Analysis","ref_index":9,"is_internal_anchor":true},{"citing_arxiv_id":"2605.11097","citing_title":"Two-loop neutrino mass model with modular $S_4$ symmetry","ref_index":126,"is_internal_anchor":false},{"citing_arxiv_id":"2605.09940","citing_title":"MSSM flavors from 7-brane configurations of magnetized SYM on $R^{1,3} \\times (T^2)^3/(Z_2 \\times Z'_2)$","ref_index":2,"is_internal_anchor":false},{"citing_arxiv_id":"2604.21979","citing_title":"Quark hierarchies and CP violation from the Siegel modular group","ref_index":4,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT","json":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT.json","graph_json":"https://pith.science/api/pith-number/LKU3DIMES3WZTYVAPX6YUFA4TT/graph.json","events_json":"https://pith.science/api/pith-number/LKU3DIMES3WZTYVAPX6YUFA4TT/events.json","paper":"https://pith.science/paper/LKU3DIME"},"agent_actions":{"view_html":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT","download_json":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT.json","view_paper":"https://pith.science/paper/LKU3DIME","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1003.3552&json=true","fetch_graph":"https://pith.science/api/pith-number/LKU3DIMES3WZTYVAPX6YUFA4TT/graph.json","fetch_events":"https://pith.science/api/pith-number/LKU3DIMES3WZTYVAPX6YUFA4TT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT/action/storage_attestation","attest_author":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT/action/author_attestation","sign_citation":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT/action/citation_signature","submit_replication":"https://pith.science/pith/LKU3DIMES3WZTYVAPX6YUFA4TT/action/replication_record"}},"created_at":"2026-05-18T02:24:36.751167+00:00","updated_at":"2026-05-18T02:24:36.751167+00:00"}